A mapping F between categories C and D that assigns to each object X of C an object F(X) of D and to each morphism f: X→Y in C a morphism F(f): F(X)→F(Y) in D, preserving identities (F(id_X)=id_{F(X)}) and composition (F(g◦f)=F(g)◦F(f)).

Principle

A functor translates objects and morphisms from one categorical context to another in a way that respects compositional structure, enabling comparison and transport of constructions.

Demonstration

The power set functor P: Set → Set assigns to each set X its power set P(X) and to each function f: X→Y the function P(f): P(X)→P(Y) given by image mapping S ↦ f(S); identities and composition are preserved.

Misapplication

Defining a map on objects and morphisms that fails to preserve composition (e.g., sending composites to non-composites) or identities invalidates functoriality and breaks induced structure like naturality conditions.

Consequence

Functors allow one to transport limits, colimits, algebraic structure, and properties between categories and to form categories of functors; they are the morphisms in the 'category of categories'.

Reversal

A contravariant functor reverses arrow direction (assigns f: X→Y to F(f): F(Y)→F(X)); forgetting variance equivalently changes which structures are preserved versus reversed and affects adjunctions.

Boundary

Excludes mappings that are only object-level or only morphism-level; enriched or lax functors relax preservation conditions and lie outside the strict functor definition unless specified.

Semantic Tension

Tension between viewing functors as mere object mappings versus structure-preserving transformations; the crucial aspect is preservation of identities and composition, not just correspondence of objects.

Synthesis

A functor F: C→D is a structure-preserving translator between categories assigning objects to objects and morphisms to morphisms while respecting identities and composition, enabling coherent transfer of categorical constructions.